Solve 1/x-1-2<0

$$\frac { 1 } { x - 1 } - 2 < 0$$

Solve for x

$$x\in \left(-\infty,1\right)\cup \left(\frac{3}{2},\infty\right)$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-1}{x-1}.

$$\frac{1}{x-1}-\frac{2\left(x-1\right)}{x-1}<0$$

Since \frac{1}{x-1} and \frac{2\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1-2\left(x-1\right)}{x-1}<0$$

Do the multiplications in 1-2\left(x-1\right).

$$\frac{1-2x+2}{x-1}<0$$

Combine like terms in 1-2x+2.

$$\frac{3-2x}{x-1}<0$$

For the quotient to be negative, 3-2x and x-1 have to be of the opposite signs. Consider the case when 3-2x is positive and x-1 is negative.

$$3-2x>0 x-1<0$$

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when x-1 is positive and 3-2x is negative.

$$x-1>0 3-2x<0$$

The solution satisfying both inequalities is x>\frac{3}{2}.

$$x>\frac{3}{2}$$

The final solution is the union of the obtained solutions.

$$x<1\text{; }x>\frac{3}{2}$$